Recoloring graphs via tree decompositions

Abstract

Let k be an integer. Two vertex k-colorings of a graph are adjacent if they differ on exactly one vertex. A graph is k-mixing if any proper k-coloring can be transformed into any other through a sequence of adjacent proper k-colorings.Jerrum proved that any graph is k-mixing if k is at least the maximum degree plus two. We first improve Jerrum’s bound using the Grundy number, which is the greatest number of colors in a greedy coloring. As a corollary, we obtain that any cograph G is (χ(G)+1)-mixing, and that for fixed χ(G) the shortest sequence between any two (χ(G)+1)-colorings is at most linear in the number of vertices. We additionally argue that while cographs are exactly the P4-free graphs, the result cannot be extended to P5-free graphs.Any graph is (tw+2)-mixing, where tw denotes the treewidth of the graph (Cereceda 2006). We prove that the shortest sequence between any two (tw+2)-colorings is at most quadratic (which is optimal up to a constant factor), a problem left open in Bonamy et al. (2012).All the proofs are constructive and lead to polynomial-time recoloring algorithms: given two colorings, we can exhibit in polynomial time a sequence transforming one into the other.